0 QTRS
↳1 DependencyPairsProof (⇔)
↳2 QDP
↳3 DependencyGraphProof (⇔)
↳4 AND
↳5 QDP
↳6 QDPOrderProof (⇔)
↳7 QDP
↳8 PisEmptyProof (⇔)
↳9 TRUE
↳10 QDP
↳11 QDPOrderProof (⇔)
↳12 QDP
↳13 PisEmptyProof (⇔)
↳14 TRUE
↳15 QDP
↳16 QDPOrderProof (⇔)
↳17 QDP
↳18 PisEmptyProof (⇔)
↳19 TRUE
↳20 QDP
↳21 QDPOrderProof (⇔)
↳22 QDP
↳23 PisEmptyProof (⇔)
↳24 TRUE
↳25 QDP
↳26 QDPOrderProof (⇔)
↳27 QDP
↳28 PisEmptyProof (⇔)
↳29 TRUE
↳30 QDP
↳31 QDPOrderProof (⇔)
↳32 QDP
↳33 PisEmptyProof (⇔)
↳34 TRUE
↳35 QDP
+(0, y) → y
+(s(x), y) → s(+(x, y))
++(nil, ys) → ys
++(:(x, xs), ys) → :(x, ++(xs, ys))
sum(:(x, nil)) → :(x, nil)
sum(:(x, :(y, xs))) → sum(:(+(x, y), xs))
sum(++(xs, :(x, :(y, ys)))) → sum(++(xs, sum(:(x, :(y, ys)))))
-(x, 0) → x
-(0, s(y)) → 0
-(s(x), s(y)) → -(x, y)
quot(0, s(y)) → 0
quot(s(x), s(y)) → s(quot(-(x, y), s(y)))
length(nil) → 0
length(:(x, xs)) → s(length(xs))
hd(:(x, xs)) → x
avg(xs) → quot(hd(sum(xs)), length(xs))
+1(s(x), y) → +1(x, y)
++1(:(x, xs), ys) → ++1(xs, ys)
SUM(:(x, :(y, xs))) → SUM(:(+(x, y), xs))
SUM(:(x, :(y, xs))) → +1(x, y)
SUM(++(xs, :(x, :(y, ys)))) → SUM(++(xs, sum(:(x, :(y, ys)))))
SUM(++(xs, :(x, :(y, ys)))) → ++1(xs, sum(:(x, :(y, ys))))
SUM(++(xs, :(x, :(y, ys)))) → SUM(:(x, :(y, ys)))
-1(s(x), s(y)) → -1(x, y)
QUOT(s(x), s(y)) → QUOT(-(x, y), s(y))
QUOT(s(x), s(y)) → -1(x, y)
LENGTH(:(x, xs)) → LENGTH(xs)
AVG(xs) → QUOT(hd(sum(xs)), length(xs))
AVG(xs) → HD(sum(xs))
AVG(xs) → SUM(xs)
AVG(xs) → LENGTH(xs)
+(0, y) → y
+(s(x), y) → s(+(x, y))
++(nil, ys) → ys
++(:(x, xs), ys) → :(x, ++(xs, ys))
sum(:(x, nil)) → :(x, nil)
sum(:(x, :(y, xs))) → sum(:(+(x, y), xs))
sum(++(xs, :(x, :(y, ys)))) → sum(++(xs, sum(:(x, :(y, ys)))))
-(x, 0) → x
-(0, s(y)) → 0
-(s(x), s(y)) → -(x, y)
quot(0, s(y)) → 0
quot(s(x), s(y)) → s(quot(-(x, y), s(y)))
length(nil) → 0
length(:(x, xs)) → s(length(xs))
hd(:(x, xs)) → x
avg(xs) → quot(hd(sum(xs)), length(xs))
LENGTH(:(x, xs)) → LENGTH(xs)
+(0, y) → y
+(s(x), y) → s(+(x, y))
++(nil, ys) → ys
++(:(x, xs), ys) → :(x, ++(xs, ys))
sum(:(x, nil)) → :(x, nil)
sum(:(x, :(y, xs))) → sum(:(+(x, y), xs))
sum(++(xs, :(x, :(y, ys)))) → sum(++(xs, sum(:(x, :(y, ys)))))
-(x, 0) → x
-(0, s(y)) → 0
-(s(x), s(y)) → -(x, y)
quot(0, s(y)) → 0
quot(s(x), s(y)) → s(quot(-(x, y), s(y)))
length(nil) → 0
length(:(x, xs)) → s(length(xs))
hd(:(x, xs)) → x
avg(xs) → quot(hd(sum(xs)), length(xs))
The following pairs can be oriented strictly and are deleted.
The remaining pairs can at least be oriented weakly.
LENGTH(:(x, xs)) → LENGTH(xs)
trivial
trivial
+(0, y) → y
+(s(x), y) → s(+(x, y))
++(nil, ys) → ys
++(:(x, xs), ys) → :(x, ++(xs, ys))
sum(:(x, nil)) → :(x, nil)
sum(:(x, :(y, xs))) → sum(:(+(x, y), xs))
sum(++(xs, :(x, :(y, ys)))) → sum(++(xs, sum(:(x, :(y, ys)))))
-(x, 0) → x
-(0, s(y)) → 0
-(s(x), s(y)) → -(x, y)
quot(0, s(y)) → 0
quot(s(x), s(y)) → s(quot(-(x, y), s(y)))
length(nil) → 0
length(:(x, xs)) → s(length(xs))
hd(:(x, xs)) → x
avg(xs) → quot(hd(sum(xs)), length(xs))
-1(s(x), s(y)) → -1(x, y)
+(0, y) → y
+(s(x), y) → s(+(x, y))
++(nil, ys) → ys
++(:(x, xs), ys) → :(x, ++(xs, ys))
sum(:(x, nil)) → :(x, nil)
sum(:(x, :(y, xs))) → sum(:(+(x, y), xs))
sum(++(xs, :(x, :(y, ys)))) → sum(++(xs, sum(:(x, :(y, ys)))))
-(x, 0) → x
-(0, s(y)) → 0
-(s(x), s(y)) → -(x, y)
quot(0, s(y)) → 0
quot(s(x), s(y)) → s(quot(-(x, y), s(y)))
length(nil) → 0
length(:(x, xs)) → s(length(xs))
hd(:(x, xs)) → x
avg(xs) → quot(hd(sum(xs)), length(xs))
The following pairs can be oriented strictly and are deleted.
The remaining pairs can at least be oriented weakly.
-1(s(x), s(y)) → -1(x, y)
trivial
trivial
+(0, y) → y
+(s(x), y) → s(+(x, y))
++(nil, ys) → ys
++(:(x, xs), ys) → :(x, ++(xs, ys))
sum(:(x, nil)) → :(x, nil)
sum(:(x, :(y, xs))) → sum(:(+(x, y), xs))
sum(++(xs, :(x, :(y, ys)))) → sum(++(xs, sum(:(x, :(y, ys)))))
-(x, 0) → x
-(0, s(y)) → 0
-(s(x), s(y)) → -(x, y)
quot(0, s(y)) → 0
quot(s(x), s(y)) → s(quot(-(x, y), s(y)))
length(nil) → 0
length(:(x, xs)) → s(length(xs))
hd(:(x, xs)) → x
avg(xs) → quot(hd(sum(xs)), length(xs))
QUOT(s(x), s(y)) → QUOT(-(x, y), s(y))
+(0, y) → y
+(s(x), y) → s(+(x, y))
++(nil, ys) → ys
++(:(x, xs), ys) → :(x, ++(xs, ys))
sum(:(x, nil)) → :(x, nil)
sum(:(x, :(y, xs))) → sum(:(+(x, y), xs))
sum(++(xs, :(x, :(y, ys)))) → sum(++(xs, sum(:(x, :(y, ys)))))
-(x, 0) → x
-(0, s(y)) → 0
-(s(x), s(y)) → -(x, y)
quot(0, s(y)) → 0
quot(s(x), s(y)) → s(quot(-(x, y), s(y)))
length(nil) → 0
length(:(x, xs)) → s(length(xs))
hd(:(x, xs)) → x
avg(xs) → quot(hd(sum(xs)), length(xs))
The following pairs can be oriented strictly and are deleted.
The remaining pairs can at least be oriented weakly.
QUOT(s(x), s(y)) → QUOT(-(x, y), s(y))
trivial
trivial
-(x, 0) → x
-(s(x), s(y)) → -(x, y)
-(0, s(y)) → 0
+(0, y) → y
+(s(x), y) → s(+(x, y))
++(nil, ys) → ys
++(:(x, xs), ys) → :(x, ++(xs, ys))
sum(:(x, nil)) → :(x, nil)
sum(:(x, :(y, xs))) → sum(:(+(x, y), xs))
sum(++(xs, :(x, :(y, ys)))) → sum(++(xs, sum(:(x, :(y, ys)))))
-(x, 0) → x
-(0, s(y)) → 0
-(s(x), s(y)) → -(x, y)
quot(0, s(y)) → 0
quot(s(x), s(y)) → s(quot(-(x, y), s(y)))
length(nil) → 0
length(:(x, xs)) → s(length(xs))
hd(:(x, xs)) → x
avg(xs) → quot(hd(sum(xs)), length(xs))
++1(:(x, xs), ys) → ++1(xs, ys)
+(0, y) → y
+(s(x), y) → s(+(x, y))
++(nil, ys) → ys
++(:(x, xs), ys) → :(x, ++(xs, ys))
sum(:(x, nil)) → :(x, nil)
sum(:(x, :(y, xs))) → sum(:(+(x, y), xs))
sum(++(xs, :(x, :(y, ys)))) → sum(++(xs, sum(:(x, :(y, ys)))))
-(x, 0) → x
-(0, s(y)) → 0
-(s(x), s(y)) → -(x, y)
quot(0, s(y)) → 0
quot(s(x), s(y)) → s(quot(-(x, y), s(y)))
length(nil) → 0
length(:(x, xs)) → s(length(xs))
hd(:(x, xs)) → x
avg(xs) → quot(hd(sum(xs)), length(xs))
The following pairs can be oriented strictly and are deleted.
The remaining pairs can at least be oriented weakly.
++1(:(x, xs), ys) → ++1(xs, ys)
trivial
trivial
+(0, y) → y
+(s(x), y) → s(+(x, y))
++(nil, ys) → ys
++(:(x, xs), ys) → :(x, ++(xs, ys))
sum(:(x, nil)) → :(x, nil)
sum(:(x, :(y, xs))) → sum(:(+(x, y), xs))
sum(++(xs, :(x, :(y, ys)))) → sum(++(xs, sum(:(x, :(y, ys)))))
-(x, 0) → x
-(0, s(y)) → 0
-(s(x), s(y)) → -(x, y)
quot(0, s(y)) → 0
quot(s(x), s(y)) → s(quot(-(x, y), s(y)))
length(nil) → 0
length(:(x, xs)) → s(length(xs))
hd(:(x, xs)) → x
avg(xs) → quot(hd(sum(xs)), length(xs))
+1(s(x), y) → +1(x, y)
+(0, y) → y
+(s(x), y) → s(+(x, y))
++(nil, ys) → ys
++(:(x, xs), ys) → :(x, ++(xs, ys))
sum(:(x, nil)) → :(x, nil)
sum(:(x, :(y, xs))) → sum(:(+(x, y), xs))
sum(++(xs, :(x, :(y, ys)))) → sum(++(xs, sum(:(x, :(y, ys)))))
-(x, 0) → x
-(0, s(y)) → 0
-(s(x), s(y)) → -(x, y)
quot(0, s(y)) → 0
quot(s(x), s(y)) → s(quot(-(x, y), s(y)))
length(nil) → 0
length(:(x, xs)) → s(length(xs))
hd(:(x, xs)) → x
avg(xs) → quot(hd(sum(xs)), length(xs))
The following pairs can be oriented strictly and are deleted.
The remaining pairs can at least be oriented weakly.
+1(s(x), y) → +1(x, y)
trivial
trivial
+(0, y) → y
+(s(x), y) → s(+(x, y))
++(nil, ys) → ys
++(:(x, xs), ys) → :(x, ++(xs, ys))
sum(:(x, nil)) → :(x, nil)
sum(:(x, :(y, xs))) → sum(:(+(x, y), xs))
sum(++(xs, :(x, :(y, ys)))) → sum(++(xs, sum(:(x, :(y, ys)))))
-(x, 0) → x
-(0, s(y)) → 0
-(s(x), s(y)) → -(x, y)
quot(0, s(y)) → 0
quot(s(x), s(y)) → s(quot(-(x, y), s(y)))
length(nil) → 0
length(:(x, xs)) → s(length(xs))
hd(:(x, xs)) → x
avg(xs) → quot(hd(sum(xs)), length(xs))
SUM(:(x, :(y, xs))) → SUM(:(+(x, y), xs))
+(0, y) → y
+(s(x), y) → s(+(x, y))
++(nil, ys) → ys
++(:(x, xs), ys) → :(x, ++(xs, ys))
sum(:(x, nil)) → :(x, nil)
sum(:(x, :(y, xs))) → sum(:(+(x, y), xs))
sum(++(xs, :(x, :(y, ys)))) → sum(++(xs, sum(:(x, :(y, ys)))))
-(x, 0) → x
-(0, s(y)) → 0
-(s(x), s(y)) → -(x, y)
quot(0, s(y)) → 0
quot(s(x), s(y)) → s(quot(-(x, y), s(y)))
length(nil) → 0
length(:(x, xs)) → s(length(xs))
hd(:(x, xs)) → x
avg(xs) → quot(hd(sum(xs)), length(xs))
The following pairs can be oriented strictly and are deleted.
The remaining pairs can at least be oriented weakly.
SUM(:(x, :(y, xs))) → SUM(:(+(x, y), xs))
trivial
trivial
+(0, y) → y
+(s(x), y) → s(+(x, y))
++(nil, ys) → ys
++(:(x, xs), ys) → :(x, ++(xs, ys))
sum(:(x, nil)) → :(x, nil)
sum(:(x, :(y, xs))) → sum(:(+(x, y), xs))
sum(++(xs, :(x, :(y, ys)))) → sum(++(xs, sum(:(x, :(y, ys)))))
-(x, 0) → x
-(0, s(y)) → 0
-(s(x), s(y)) → -(x, y)
quot(0, s(y)) → 0
quot(s(x), s(y)) → s(quot(-(x, y), s(y)))
length(nil) → 0
length(:(x, xs)) → s(length(xs))
hd(:(x, xs)) → x
avg(xs) → quot(hd(sum(xs)), length(xs))
SUM(++(xs, :(x, :(y, ys)))) → SUM(++(xs, sum(:(x, :(y, ys)))))
+(0, y) → y
+(s(x), y) → s(+(x, y))
++(nil, ys) → ys
++(:(x, xs), ys) → :(x, ++(xs, ys))
sum(:(x, nil)) → :(x, nil)
sum(:(x, :(y, xs))) → sum(:(+(x, y), xs))
sum(++(xs, :(x, :(y, ys)))) → sum(++(xs, sum(:(x, :(y, ys)))))
-(x, 0) → x
-(0, s(y)) → 0
-(s(x), s(y)) → -(x, y)
quot(0, s(y)) → 0
quot(s(x), s(y)) → s(quot(-(x, y), s(y)))
length(nil) → 0
length(:(x, xs)) → s(length(xs))
hd(:(x, xs)) → x
avg(xs) → quot(hd(sum(xs)), length(xs))